Abstract

A standard tool for classifying the complexity of equivalence relations on \(\omega\) is provided by computable reducibility. This reducibility gives rise to a rich degree structure. The paper studies equivalence relations, which induce minimal degrees with respect to computable reducibility. Let \(\Gamma\) be one of the following classes: \(\Sigma^{0}_{\alpha}\), \(\Pi^{0}_{\alpha}\), \(\Sigma^{1}_{n}\), or \(\Pi^{1}_{n}\), where \(\alpha\geq 2\) is a computable ordinal and \(n\) is a non-zero natural number. We prove that there are infinitely many pairwise incomparable minimal equivalence relations that are properly in \(\Gamma\).

中文翻译：

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超算术和分析层次结构中的最小等价关系

摘要

可计算可简化性提供了一种用于对\（\ omega \）上等价关系的复杂性进行分类的标准工具。这种还原性产生了丰富的度结构。本文研究了等价关系，这些等价关系在可计算可约性方面引入了最小程度。假设\（\ Gamma \）为以下类别之一：\（\ Sigma ^ {0} _ {\ alpha} \），\（\ Pi ^ {0} _ {\ alpha} \），\（\ Sigma ^ {1} _ {n} \）或\（\ Pi ^ {1} _ {n} \），其中\（\ alpha \ geq 2 \）是可计算序数，而\（n \）是非序数-零自然数。我们证明存在无限多的成对不可比的最小等价关系\（\ Gamma \）。